Mathematical and statistical models are important tools in the design, analysis and interpretation of experiments in toxicology and biochemistry. However, many of the properties of the existing models are poorly understood. In addition, many of the existing models do not adequately describe current research findings in these areas. The goal of this project is to increase our understanding of the use and application of existing models in toxicology and biochemistry and to develop and implement new models to aid in explaining new research findings. One research effort concerns the use of physiologically based pharmacokinetic models in estimating carcinogenic risk from exposure to chemicals. A method was developed for incorporating all the sources of variability into the risk estimation procedure to arrive at an estimate of the overall variability. It was shown that the use of these models was likely to increase the variability of the risk estimates. In another study, the ability of carcinogenicity data to estimate parameters in a clonal two-stage model of carcinogenesis was examined. The findings indicated that the carcinogenicity data was inadequate for this purpose and additional information must be used. Research on this topic is continuing with a focus on redesigning carcinogenicity experiments to provide better information on these model parameters. Other research topics concern correcting for survival differences in animal carcinogenicity data and the potential risks from using inappropriate models in estimating risks of developmental defects from exposure to chemicals.